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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic properties of the residual bootstrap for Lasso estimators


Authors: A. Chatterjee and S. N. Lahiri
Journal: Proc. Amer. Math. Soc. 138 (2010), 4497-4509
MSC (2010): Primary 62J07; Secondary 62G09, 62E20
Published electronically: July 9, 2010
MathSciNet review: 2680074
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Abstract: In this article, we derive the asymptotic distribution of the bootstrapped Lasso estimator of the regression parameter in a multiple linear regression model. It is shown that under some mild regularity conditions on the design vectors and the regularization parameter, the bootstrap approximation converges weakly to a random measure. The convergence result rigorously establishes a previously known heuristic formula for the limit distribution of the bootstrapped Lasso estimator. It is also shown that when one or more components of the regression parameter vector are zero, the bootstrap may fail to be consistent.


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Additional Information

A. Chatterjee
Affiliation: Department of Statistics, Texas A&M University, College Station, Texas 77843-3143
Email: cha@stat.tamu.edu

S. N. Lahiri
Affiliation: Department of Statistics, Texas A&M University, College Station, Texas 77843-3143
Email: snlahiri@stat.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10474-4
PII: S 0002-9939(2010)10474-4
Keywords: Consistency, bootstrap, penalized regression, random measure
Received by editor(s): January 22, 2009
Received by editor(s) in revised form: December 23, 2009, and March 2, 2010
Published electronically: July 9, 2010
Additional Notes: This research was partially supported by NSF grant DMS-0707139.
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2010 American Mathematical Society